The axial anomaly and the phases of dense QCD Gordon Baym University of Illinois In collaboration with Tetsuo Hatsuda, Motoi Tachibana, & Naoki Yamamoto Quark Matter 2008 Jaipur 6 February 2008 Title

Color superconductivity

Color superconductivity

Phase diagram of equilibrated quark gluon plasma Critical point Asakawa-Yazaki 1989. 1st order crossover Karsch & Laermann, 2003

New critical point in phase diagram: induced by chiral condensate – diquark pairing coupling via axial anomaly Hatsuda, Tachibana, Yamamoto & GB, PRL 97, 122001 (2006) Yamamoto, Hatsuda, Tachibana & GB, PRD76, 074001 (2007) Hadronic Normal QGP Color SC (as ms increases)

Order parameters In hadronic (NG) phase: = color singlet chiral field a,b,c = color i,j,k = flavor C: charge conjugation In hadronic (NG) phase: = color singlet chiral field In color superconducting phase : U(1)A axial anomaly => Coupling via ‘t Hooft 6-quark interaction » dR     » 3 dLy » dLy dR

Ginzburg-Landau approach In neighborhood of transitions, d (pair field) and  (chiral field) are small. Expand free energy  (cf. with free energy for d =  = 0) in powers of d and : = chiral + pairing + chiral-pairing interactions

Chiral free energy (from anomaly) m02» 3 Pisarski & Wilczek, 1984 m02» 3 (from anomaly) a0 becoming negative => 2nd order transition to broken chiral symmetry

Quark BCS pairing (diquark) free energy (Iida & GB 2001) Transition to color superconductivity when 0 becomes negative d fully invariant under: G = SU(3)L×SU(3)R×U(1)B×U(1)A×SU(3)C

G = SU(3)L×SU(3)R×U(1)B×U(1)A×SU(3)C Chiral-diquark coupling: (to fourth order in the fields) tr over flavor Leading term ("triple boson" coupling) » 1 arises from axial anomaly. Pairing fields generate mass for chiral field.  terms invariant under: G = SU(3)L×SU(3)R×U(1)B×U(1)A×SU(3)C

Three massless flavors Simplest assumption: Color-flavor locking (CFL) Alford, Rajagopal & Wilczek (1998) then c and  terms arise from the anomaly. ‘t Hooft interaction =>  has same sign as c (>0) and similar magnitude From microscopic computations (weak-coupling QCD, NJL) ¿ 1 If b < 0, need 6 f-term to stabilize system.

Warm-up problem: first ignore -d couplings : ==0, b>0 => 1st order chiral transition 2nd order pairing transition Hadronic (NG)  ≠ 0, d=0 Normal (NOR)  = d=0 T NG= Nambu-Goldstone NOR 2nd order NG CSC COE Coexistence (COE)  ≠ 0, d ≠ 0 Color sup (CSC)  = 0, d ≠ 0 μ Schematic phase diagram 1st order

Major modification of phase diagram via chiral-diquark interplay! Full G-L free energy with chiral-diquark coupling ( > 0,  ≥ 0) Locate phase boundaries and order of transitions by comparing free energies:  > 0,  = 0 b > 0, f = 0 no -d coupling ( =  = 0) A= new critical point Major modification of phase diagram via chiral-diquark interplay! Non-zero  <<  produces no qualitative changes b < 0 with f > 0 => qualitatively similar results

Critical point arises because d2, in -d2 term, acts as external field for , washing out 1st order transition for large d2 -- as in magnetic system in external field. With axial anomaly, NG-like and CSC-like coexistence phases have same symmetry, allowing crossover. NG and COE phases realize U(1)B differently and boundary is sharp.

Two massless flavors Assume 2-flavor CSC phase (2SC) then (Nf= 2 GL parameters / Nf=3 parameters) No cubic terms; cf. three flavors: tetracritical pt. bicritical point

Phase structure in T vs.  Mapping the phase diagram from the (a, α) plane to the (T, μ) plane requires dynamical picture to calculate G-L parameters. T  COE CSC Hadronic NG QGP No anomaly-induced critical point for Nf=2 in SU(3)C or SU(2)C T  COE (NG-like) (CSC-like) Hadronic NG QGP “Hadron”-quark continuity at low T (Schäfer-Wilczek 1999)

Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark without anomaly

Schematic phase structure of dense QCD with two light u,d quarks and a medium heavy s quark with anomaly New critical point

Finding precise location of new critical point requires phenomenological models, and lattice QCD simulation. Too cold to be accessible experimentally. To make schematic phase diagram more realistic should include * realistic quark masses * for neutron stars, charge neutrality and beta equilibrium * interplay with confinement (characterize by Polyakov loop) [e.g., R. Pisarski, PRD62 (2000); K. Fukushima, PLB591 (2004); C.Ratti, M. Thaler, W. Weise PRD73 (2006); C.Ratti, S. Rössner and W. Weise, PRD (2007) hep-ph/0609281 ]. Delineate nature of NG-like coexistence phase. * thermal gluon fluctuations * possible spatial inhomogeneities (FFLO states)

Hadron-quark matter deconfinement transition vs. BEC-BCS crossover in cold atomic fermion systems In trapped atoms continuously transform from molecules to Cooper pairs: D.M. Eagles (1969) ; A.J. Leggett, J. Phys. (Paris) C7, 19 (1980); P. Nozières and S. Schmitt-Rink, J. Low Temp Phys. 59, 195 (1985) Pairs shrink 6Li Tc/Tf » 0.2 Tc /Tf » e-1/kfa

Phase diagram of cold fermions vs. interaction strength BCS BEC of di-fermion molecules Temperature Tc Free fermions +di-fermion molecules Free fermions -1/kf a a>0 a<0 Tc/EF» 0.23 Tc» EFe-p/2kF|a| (magnetic field B) Unitary regime (Feshbach resonance) -- crossover No phase transition through crossover

Deconfinement transition vs. BEC-BCS crossover Tc free fermions molecules BCS Abuki, Itakura & Hatsuda, PRD65, 2002 BCS paired quark matter BCS-BEC crossover Hadrons Hadronic Normal Color SC BCS Possible structure of crossover (Fukushima 2004 ) In SU(2)C : Hadrons <=> 2 fermion molecules. Paired deconfined phase <=> BCS paired fermions B

Quark matter cores in neutron stars Canonical picture: compare calculations of eqs. of state of hadronic matter and quark matter. Crossing of thermodynamic potentials => first order phase transition. ex. nuclear matter using 2 & 3 body interactions, vs. pert. expansion or bag models. Akmal, Pandharipande, Ravenhall 1998 Typically conclude transition at »10nm -- not reached in neutron stars if high mass neutron stars (M>1.8M¯) are observed (e.g., Vela X-1, Cyg X-2) => no quark matter cores

More realistically, expect gradual onset of quark degrees of freedom in dense matter Hadronic Normal Color SC New critical point suggests transition to quark matter is a crossover at low T Consistent with percolation picture, that as nucleons begin to overlap, quarks percolate [GB, Physica (1979)] : nperc » 0.34 (3/4 rn3) fm-3 Quarks can still be bound even if deconfined. Calculation of equation of state remains a challenge for theorists

Continuity of pionic excitations with increasing density Low  pseudoscalar octet (,K,) goes continuously to high  diquark pseudoscalar. Octet hadron-quark continuity in excited states as well. T Quark-Gluon Plasma Gell-Mann-Oakes-Renner (GOR) relation Alford, Rajagopal, & Wilczek, 1999 Hadrons Color superconductivity ? mB Mass spectrum and form of pions at intermediate density?

Ginzburg-Landau effective Lagrangian » Pion  at low density Generalized pion  at high density Under SU(3)R,L and Axial anomaly couples  to  and to quark masses, mq : to O(M) »

Generalized pion mass spectrum Mass eigenstates: = mixed state of & with mixing angle . Generalized Gell-Mann-Oakes-Renner relation Axial anomaly（breaking U(1)A） at very high density Hadron-quark continuity also in excited states Axial anomaly plays crucial role in pion mass spectrum

Pion mass splitting unstable

Conclusion Phase structure of dense quark matter Intriguing interplay of chiral and diquark condensates U(1)A axial anomaly in 3 flavor massless quark matter => new low temperature critical point in phase structure of QCD at finite  Collective modes in intermediate density Concrete realization of quark-hadron continuity Effective field theory at moderate density => pion as generalized meson; generalized GOR relation Vector mesons, nucleons and other heavy excitations (Hatsuda, Tachibana, & Yamamoto, in preparation) Vector meson continuity

THE END

Toy Model Two complex scalar fields: Lagrangian: light “pion” heavy “pion” Diagonalize to find mass relations:

Continuous crossover from NG to CSC phases allowed by symmetry In CFL phase: dLdRy breaks chiral symmetry but preserves Z4 discrete subgroup of U(1)A. For  = 0, different symmetry breaking in two phases.  term has Z6 symmetry, with Z2 as subgroup. With axial anomaly, NG and CSC-like coexistence phases have same symmetry, and can be continuously connected. NG and COE phases realize U(1)B differently and boundary is not smoothed out. In COE phase:  breaks chiral symmetry, preserving only Z2 .